Economics 214 Lecture 15 Differential Calculus

Need for Differential Calculus We have seen the contribution of Comparative statics to our understanding of economics. We have learned to do comparative statics for linear models. Differential calculus will permit us to do comparative statics for nonlinear models.

Functions and Rates of Change Assume we have an excise tax on a good, Q. The tax per unit sold is “t”. Our revenue from the tax is R=tQ. We ask ourselves the question “How much revenue will we gain if we raise the tax from t 1 to t 2 ?” If there is no change in sales, revenue will be t 2 Q. This makes revenue a linear function of the tax rate (see line OL on figure 6.1). Increasing the excise tax, effectively increases the price of the good being sold. Law of Demand suggests that when price increases, the quantity sold will decrease. If this is true then the revenue from the excise tax after the rate increase will be less t 2 Q. In other words,  R  Q  t or the average rate of change  R/  t  Q. Our tax revenue function in general will increase at a decreasing rate and may max out at some tax rate. If this is true, our average rate or change or secant function will be different depending on the initial and final tax rate.

Figure 6.1 Tax Revenues and Tax Rates

Example of constant elasticity demand function

Graph of Demand and tax

Graph of tax revenue function for constant elasticity demand